Theorem rexv 2818

Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
rexv	|- ( E. x e. _V ph <-> E. x ph )

Proof of Theorem rexv

Step	Hyp	Ref	Expression
1		df-rex 2514	. 2 |- ( E. x e. _V ph <-> E. x ( x e. _V /\ ph ) )
2		vex 2802	. . . 4 |- x e. _V
3	2	biantrur 303	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	exbii 1651	. 2 |- ( E. x ph <-> E. x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 187	1 |- ( E. x e. _V ph <-> E. x ph )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   E.wrex 2509   _Vcvv 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-rex 2514  df-v 2801

This theorem is referenced by:  rexcom4  2823  spesbc  3115  abnex  4538  dfco2  5228  dfco2a  5229  finacn  7386